Question

49 how many terms are common in the two arithmetic progressions given below? Ap1: 1, 4, 7, ..., t45 ap2: 1, 3, 5, 7, ..., t55

Answer 1

see 1st Arithmetic progression,
1 , 4 , 7, 10, 13, 16, ..... $t_{45}$
we know, $t_n=a+(n-1)d$
so, $t_{45}=1+(45-1)\times3$
= 1 + 44 × 3 = 133
so, 1, 4, 10, 13, 16, ..... 133


and 2nd Arithmetic progression,
1, 3, 5, 7, 9, 11, 13, .... $t_{55}$
$t_{55}=1+(55-1)\times2=109$
so, 1, 3, 5, 7, 9, 11, ..... 109

here common terms are
1, 7, 13, 19, 25 ......
let's check 109 is in 1st series
109 = 1 + (n - 1) × 3
108 = 3 × (n - 1)
36 + 1 = n (integer)
hence, 109 exists in 1st series

now find number of terms in 1, 7, 13, 19, 25 ,...109
first term , a = 1, common difference , d = 6
then, 109 = 1 + (n - 1) × 6
108 = 6(n - 1)
18 = n - 1
n = 19

hence, answer is 19